Indications of de Sitter spacetime from classical sequential growth dynamics of causal sets

A large class of the dynamical laws for causal sets described by a classical process of sequential growth yields a cyclic universe, whose cycles of expansion and contraction are punctuated by single “origin elements” of the causal set. We present evidence that the effective dynamics of the immediate future of one of these origin elements, within the context of the sequential growth dynamics, yields an initial period of de Sitter-like exponential expansion, and argue that the resulting picture has many attractive features as a model of the early universe, with the potential to solve some of the standard model puzzles without any fine-tuning.

Figure 1

Mean population of levels for an $N=16384$ element random tree. The mean and its error are measured from 100 samples of the random tree process. The fitting function proportional to a Poisson distribution is shown. Here $A=16687\pm 85$ and $\lambda =9.306\pm 0.022$.

Figure 2

A sample causal set generated by originary percolation with $N=16$, $p=0.2$. Elements of the initial tree are shown by red squares.

Figure 3

Mean “spatial volume” of ten originary percolated causal sets with $N=11585$, $p=0.001$. Shown is the cardinality of level $t$ in red, the cardinality of an inextendible antichain containing level $t$ in green, the number of elements of the initial tree at level $t$ in blue, and the number of maximal elements of the initial tree, at level $t$, in magenta.

Figure 4

An Alexandrov neighborhood (order interval) in de Sitter space.

Figure 5

The set of all pairs $(L,N_{\diamond })$ for each related pair of elements, in three causal sets. Each causet was generated with the same value of $p=0.025$ but three different values of $N=500$, 1500, and 2500. (To make the figure size manageable, we plot only every 4th point for $N=500$, every 10th for $N=1500$, and every 200th point for $N=2500$.) Note that the smaller data sets are a subset of the larger ones, and that at some point the maximum $N_{\diamond }(L)$ no longer increases with $N$.

Figure 6

A plot of the maximum values of $N_{\diamond }$ as a function of $L$ for $p=0.001$ and $N=15000$, along with best fit curves for de Sitter space of three different dimensions. The vertical line on the left marks the end of the tree era, while the one on the right separates the points that “see” the finiteness of the causal set. The overall best fit is achieved from the curve for $3+1$ dimensions, and is shown in black.

Figure 7

A plot of the maximum values of $N_{\diamond }$ as a function of $L$ for $p=0.0001$ and $N=50000$, along with best fit curves for de Sitter space of three different dimensions. The overall best fit is achieved from the curve for $3+1$ dimensions.

Figure 8

A plot of the maximum values of $N_{\diamond }$ as a function of $L$ for $p=0.0002$ and $N=30000$, in log scale, along with best fit curves for de Sitter space of three different dimensions. Again the overall best fit is achieved from the curve for $3+1$ dimensions.

Figure 9

A plot of the maximum values of $N_{\diamond }$ as a function of $L$ for $p=0.01$ and $N=2000$, along with best fit curves for de Sitter space of three different dimensions. Here the overall best fit is achieved from the curve for $2+1$ dimensions.

Figure 10

Comparing behavior of the mean $N_{\diamond }$ versus the maximum. The results come from causal sets generated with $p=0.0008$, $N=15000$.